\[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {e x} \left (c+d x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {\left (-a d +b c \right )^{2} \sqrt {e x}}{c \,d^{2} e \sqrt {d \,x^{2}+c}}+\frac {2 b^{2} \sqrt {e x}\, \sqrt {d \,x^{2}+c}}{3 d^{2} e}-\frac {\left (-3 a^{2} d^{2}-6 a b c d +5 b^{2} c^{2}\right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {c}+x \sqrt {d}\right ) \sqrt {\frac {d \,x^{2}+c}{\left (\sqrt {c}+x \sqrt {d}\right )^{2}}}}{6 \cos \left (2 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right ) c^{\frac {5}{4}} d^{\frac {9}{4}} \sqrt {e}\, \sqrt {d \,x^{2}+c}} \]
command
integrate((b*x^2+a)^2/(d*x^2+c)^(3/2)/(e*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {{\left ({\left (5 \, b^{2} c^{3} - 6 \, a b c^{2} d - 3 \, a^{2} c d^{2} + {\left (5 \, b^{2} c^{2} d - 6 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt {d} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) - {\left (2 \, b^{2} c d^{2} x^{2} + 5 \, b^{2} c^{2} d - 6 \, a b c d^{2} + 3 \, a^{2} d^{3}\right )} \sqrt {d x^{2} + c} \sqrt {x}\right )} e^{\left (-\frac {1}{2}\right )}}{3 \, {\left (c d^{4} x^{2} + c^{2} d^{3}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{d^{2} e x^{5} + 2 \, c d e x^{3} + c^{2} e x}, x\right ) \]